Complexity of Regex Crosswords.

In a typical regex crossword puzzle, one is given two non-empty lists 〈R1,…,Rm〉 and 〈C1,…,Cn〉 of regular expressions (regexes) over some alphabet, and the challenge is to fill in an m×n grid of characters such that the string formed by the ith row is in L(Ri) and the string formed by the jth column is in L(Cj), for all 1≤i≤m and 1≤j≤n. We consider a restriction of this puzzle where all the Ri are equal to one another and similarly the Cj. We consider a 2-player version of this puzzle, showing it to be PSPACE-complete. Using a reduction from 3SAT, we also give a new, simple proof of the known result that the existence problem of a solution for the restricted (1-player) puzzle is NP-complete. We also prove NP-completeness where all the row and column regular expressions are equal to each other.